Problem with bridge connecting to edges of face instead of round extruded part
You have 3 problems:

Overlapping vertices on the bottom and top of the cup handle, so select all and press AltM to merge:



Inner face on the top of the cup handle, it makes the bridge bad, so first delete this face loop:



Then delete this face:



Now bridge the top and bottom of the handle with the cup, use CtrlE &gt; Bridge Edge Loops instead of LoopTools to have a good result right away.

The result is not good, it's because the normals of the handle are flipped as you see if you enable the Face Orientation option, so select the handle and press AltN to recalculate: